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Weighted Networks Model Based on Traffic Dynamics with Local Perturbation

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Weighted Networks Model Based on Traffic Dynamics with Local Perturbationon,Model,Based,with,Local,model,based,local

    Weighted Networks Model Based on Traffic

    Dynamics with Local Perturbation Commun.Theor.Phys.(Bering,China)48(2007)PP.953956

    ?InternationalAcademicPublishersv01.48.No.5.November15.2007

    WeightedNetworksModelBasedonTrafficDynamicswithLocalPerturbation ZHA0HuiandGAOZiYou

    StateKeyLaboratoryofRailTrafficControlandSafety,SchoolofTrafficandTransportation,BeijingJiaotongUniversity

    Beijing100044,China

    (ReceivedOctober30,2006)

    AbstractInthestudyofweightedcomplexnetworks,the

    muchattention.However,thevariationoftopologyandweight

    interplaybetweentrafficandtopologyhavebeenpaid

    broughtbynewaddedverticesoredgesshouldalsobe

    c0nsidered.InthisDaper,anevolutionmodelofweightednetworksdrivenbytrafficdynamicswithlocalperturbationis

    proposed.Themodelgivespower

    lawdistributionofdegree,weightandstrength,asconfirmedbempiricalmeasure

    ments.BychoosingappropriateparametersWand5,theexponentsofvariouspowerlawdistributionscanbeadjusted

    t0meetrealworldnetworks.Nontrivialclusteringcoe

    cientC,degreeassortativitycoefficient,andstrengthdegree

    correlati0narealsoconsidered.Whatshouldbeemphasizedisthat,withtheconsiderationoflocalperturbation,one

    canadjusttheexponentofstrength

    degreecorrelationmoreeffectively.Itmakesourmodelmoregeneralthanprevious onesandmayheDreproducingrealworldnetworksmoreappropriate

PACSnumbers:87.23.Kg,89.75

    Keywords:weightednetworks,

    Da,89.75.Fb,89.75.Hc

    trafficdynamics,scalefree,perturbation

    Duringthepastfewyears,manyresearchershave beendevotedtocharacterizingandunderstandingthein

    frastructuresofcomplexnetworks.Prototypicalexam

    plescoverasdiverseasthetechnologicalnetworks(Inter

    net,phonecallsnetwork),biologicalsystems(foodwebs, metabolicsystems1,andsocialnetworks(coauthorship,

    citationnetworks1.[1-4JAllofthesenetworksgenerally exhibitcomplextopologicalproperties.Whatshouldbe highlightedisthesmallworldphenomenon[5Jandscale-

    freebehavior.[1,6]SinceBarabgsiandAlbert[6Jproposed theclassicalnlodelfBAmodel1,whichintroducedgrowth andpreferentialattachmentmechanismstomimicthe largefluctuationsintheconnectivitypatternofcomplex networks,manyworkshavebeentriggeredtoresearchof scalefreenetworksandmuchprogresshasbeenachieved. However.realworldnetworksarefarfrOli3.thebinary state.Thepurelytopologicaldescription,naymissmany importantfeaturesofrealsystems.Forexamplethe trafficonInternetisfundamentalinfo?rmationtoaccess thedescriptionofthesystem.Whatismore.including thenumberofpassengersinthe,v0ridAirportNetwork, thestrengthofPredatorPreyInteractioninecological

    networks,theflowinMetabolicReactorNetworks,the numberofcoauthoredpapersinScientificCollaboration Network.allofthesystemsaregoodexamplesaswel1. Thiscallsforthemodelingapproachtocomplexnetworks

    thatgoesbeyondthebooleanstructure.Theresearchof weightednetworkshasbeenhinderedmostlybythelack ofdatafrOli3.realworld.Mostrecently.theavailability oftheempiricaldatamakesitpossibletoconsidervari

    ationofweightcouplingwiththetopologythatreflects thephysicalcharacteristicsofrealnetworks.InBarrat etnf.._7_theauthorshavemadeacomprehensiveanalysis ontheweightednetworksfrOli3.realworldsystemsin

    cludingtheScientificCoauthorshipNetworkand,v0rid AirportNetwork.Then,manymodelswereproposedto mimictheweightedcomplexnetworks.[8--12JWhatshould behighlightedistheworksbyBarrateta1.._9_whichpre

    senteda,nodelfBBVmodel1thatintegratesthetopol

    ogyandtheweightdynamicstostudytheevolutionof weightednetworks.BasedonthetraLfficdrivenmecha- nism.,vangeta1.0Jproposedanother,nodeltocharac

    terizetheweightedtechnologicalnetworks. Inouropinion,thetrafficdrivenmechanismandthe variationbroughtbythenewaddedverticesoredges shouldbeconsideredsynthetically.Inthispaper,wepro

    poseatrafficdriven,nodelfortheweightednetworkswith localperturbation.Inourmode1.verticesenterthesys

    ternconstantlyandnewedgesareallowedtobeaddedbe

    tweenoldvertices.Thepreexistingtrafficflowalongthe

    linksupdateswiththegrowthofnetworks.Furthermore. theadditionofnewverticescouldperturbtheweightof existingedgeslocally.Theemergenceofedgesbetweenal

    readyexistingverticesandthevariationofweightbrought bynewaddedverticesoredges113.ayallcontributetothe growthofthenetworks.Infact,thephenomena113.ayex-

istinmanytransDortationnetworksincludingthe?brid

    AirportNetwork,Internet,etc.Theevolutionprocessof ,v0ridAirportNetworkismostlydrivenbytheinternal increasingtrafficdemand.Andthelocalperturbationis alsoreasonable.Actually,whenanewairlineisconnected, itwillgenerallymodirythetrafficactivitybetweenairport anditsneighboringairports.Passengersbroughtbythe newconnectionwillincreasethepassengerflowonthe otherroutes.ConsideringtheInternetaswell,theneed ofaredundancywiringandanincreasingneedofavail

    ablebandwidthfordatatransmissionisaninherentdriven mechanism.Anditisalsoreasonabletorealizethatthe newconnectiontoanewrouterwillincreasethetrafficon otherroute'slinks.

    ,ightednetworksareoftendescribedbyanadjacency matrixwid,whichdenotestheweightontheedgeconnect

    ingverticesiandJ,withi,J=1,...,N,whereNisthe TheprojectsupportedbyNationalNaturalScienceFoundationofChinaunderGrantNo.706

    31001,ChaniangScholarsandInnovative

    ResearchTeaminUniversityunderGrantNo.IRT0605,andtheStateKeyBasicResearchPro

    gramofChinaunderGrantNo2006CB705500

954ZHAoHuiandGAOZiY0uV0l_48

    sizeofthenetwork,(wij=0ifthenodesiandJare notadjacent).Ageneralizationofdegreeinthecaseof weightednetworksisthestrengthofvertex[7,s]described aS

    s

    ?wiJ,

    wherethesumrunsoverthesetV(i)ofneighborsofnode

i.Thestrengthofavertexhastheinformationinclud

    ingitsconnectivityandtheweightsofitslinks.Proto- typicalcasesofcomplexnetworksusuallyexhibitpower lawdegreedistributionP(>k)I?,[71l.]strength

    distributionPf>s18-[7]andweightdistribution P(>w)w).[14]Highlycorrelatedwiththedegree, thestrengthoftendisplaysscalefreepropertysk

    too.[14,15

    ThemodelstartsfromaninitialconfigurationofNo verticesfullyconnectedbylinkswithassignedweightw0. Theevolutionprocesscanbedefinedonthreemecha- nisms:topologicalgrowth,trafficdrivendynamicsand localperturbation.

    (i)Topologicalgrowth.Ateachtimestep,anewver

    rexwithmedgesconnectedisaddedtompreexisting vertices,preferentiallychoosingnodeswithlargestrength ?new+8i(1)

    Thismechanismiswidelyusedasangeneralizationof degreepreferentialattachment[9,10]andisreasonablein modelingweightedcomplexnetworks.

    (ii)Trafficdynamics.Fromthebeginningoftheevo

    lution.allpossibleconnectionsupdatetheirweightsac

    cordingtothesocalledstrengthcouplingmechanism[J

    ateachtimestep:

    训巧【训wij

    ,

    +

    w

    w

    i

it

    th

    hp

    prob rob

    a

    a

    b

    b

    i

    i

    l

    l

    i

    i

    t

    t

    y

    yW

    1

    pij,

    where p

    8i8j integratesthestrengthcouplingofverticesiandJ,and

    determinestheincrementprobabilityofweightij(ifi

    andareunconnected,wj=0).Inthisstep,thetotal

    weightoftheedgesinstatisticalsenseismodifiedbythe

    amount(?t<{Awij)=W.Thealwaysgrowingtraffic playsthedrivingroleinnetworkevolution.Infact,Wpij isverylikelytoexceedoneiftheinitialnumberofnodes Noissmal1.Inourmodel,whenjexceedsone,itis automaticallyassumedtobeone.

    isproportionallydistributedamongtheedgesattachedto thevertexaccordingtotheirweights.

    Theevolutiontimeismeasuredwithrespecttothe numberofnodesaddedtothegraph,thatist=N一?0,

    andthenaturaltimescaleofthemodeldynamicsis thenetworksizeN.Usingthecontinuousapproxima- tion,wecantreatk,w,s,andthetimetascontinuous variables.[4,9]theevolutioncanbestudiedanalyticallyby inspectthetimeevolutionoftheaveragevaluesofsand tofthei-thvertexattimet.Theweightwijincreases bythetrafficdrivenmechanism(2)andtheadditionofa newlink(eitheronioronJ)(4).Sothecorresponding rateequationcanbewrittenas

    

    dwij

    :+mdt?n,6fn?6)8a8b一?fSl8i

    +mL(~wij.

    .

    (6)

    zslsj

    Anypossible(existingornot)connectionstoiareup

    datedbythetrafficdrivendynamicsbringsincrementof itsstrength.Whatismore,whenanewedgeisaddedto thenetwork,thestrength8iofvertexicanincrease,ifthe edgeconnectsdirectlytoiortooneofitsneighbors,

    ?j(?t)2Wsisj

    dt一?.s.?b(?.)sbsfsi

    +m

    Bynoticing?si(t)=2(m(1+)+W)t,onecananalyt

    icallyobtainthestrengthdistributionP(s1swith

    theexponent.[,.]

    =2+

    2W+m+2m5

    Obviously,whenW=0and=0theexponentis-y=3 anditisequivalenttotheclassicalBAnetwork.Forthe increasingvaluesofWor,thedistributionisgradually broaderwith-ydecreasesapproaching2. 1()()

    1【】

    a_

    fiii1Localperturbation.Thepresenceofthenew edge(,i)willintroducevariationsoftheexistingweights10z

    acrossthenetwork.Hereweconsiderthelocalrearrange

    mentsofweightsbetweenianditsneighborsJ?V(i)

    accordingtothesimplerule,[.]

    wij_?wij+Awij

    where

    ?训ii:(~wij.

    .

    St

    Thatmeanstheintroductionofanewedgeofweightw0 withthenewvertexinducesanincreaseoftrafficthat 1【】(1

1【】2

    Fig.1Cumulativestrengthdistributionforvariouspa

    rametersWand5byfixing(a):W=1and(b):5=1. TheplotisconsistentwithscalefreebehaviorP(>s)^

    s

    .

    Lineardatafittinggivestheslopeconsistenttothe analyticalprediction=2+m/(2w+m+2m5).The dataareaveragedover20networkswithsizeN=2000 evolvedindependently.

    ?,m+

    ??

No.5WeightedNetworksModelBasedonTrafficDynamicswithLocalPerturbation955

    Inordertochecktheanalyticalpredictions.weper

    formednumericalsimulationsextensively.Networksare generatedbychoosingdifierentvaluesofWand5,with No=3m=3andwn=1fixed.Infactonecanfindthat thevariousscale-freepropertiesofourmodelnetworksare almostindependentoftheinitialconfigurations. 10o

    1()2

    

    1(()

    l()1

    1()2

    1(

    Fig.2Cumulativedegreedistributionforvariouspa

    rametersWand5byfixing(a):5=1and(b):W=1.

ThepowerlawdistributionP(k1^kcanbeob

    served.Thedataareaveragedover20networkswith sizeN=2000evolvedindependently.

    102

    1()3

    1()4

    _

    1()

    103

    1(4

    l()5

    106

    1(1

    10

    Fig.3Cumulativeweightdistributionforvariouspa

    rametersWandbyfixing(a):=1and(b):W=1. ThepowerlawweightdistributionP(w1^一)canbe

    checked.Thedataareaveragedover20independently evolvednetworkswithsizeN=2000.

    ThecumulativestrengthdistributionPf>s1ofthe modelnetworkswithvariousparametersWand5isgiven inFig.1.Asisillustrated.thestrengthdistributionfo1. 1owsapowerlawP(s)s1,whichisingoodagreement

    withthetheoreticalpredictions.PromFig.2.thescale- freepropertyofdegreedistributionPf1"canbe

    checkedwithdifferentWand5.Asisreported,both Wand5mayaffecttheexponentofthedegreedistribu

    tion.Comparingwiththetracdrivenmechanism.one

    canfindthatthelocalperturbationcontributeslessto thevariationoftheexponentofdegreedistribution.The

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